Geodesic
More tales of Hoffman: bounds for the vector chromatic number of a graph
Discussiones Mathematicae. Graph Theory, Tome 43 (2023) no. 1, pp. 159-169

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Let χ(G) denote the chromatic number of a graph and χ_v(G) denote the vector chromatic number. For all graphs χ_v(G) ≤χ(G) and for some graphs χ_v(G) ≪χ(G). Galtman proved that Hoffman's well-known lower bound for χ(G) is in fact a lower bound for χ_v(G). We prove that two more spectral lower bounds for χ(G) are also lower bounds for χ_v(G). We then use one of these bounds to derive a new characterization of χ_v(G).
Keywords: vector chromatic number, spectral bounds 
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Wocjan, Pawel; Elphick, Clive; Anekstein, David. More tales of Hoffman: bounds for the vector chromatic number of a graph. Discussiones Mathematicae. Graph Theory, Tome 43 (2023) no. 1, pp. 159-169. http://geodesic.mathdoc.fr/item/DMGT_2023_43_1_a9/